Generalized linear array model
In statistics, the generalized linear array model(GLAM) is used for analyzing data sets with array structures. It based on the generalized linear model with the design matrix written as a Kronecker product.
The generalized linear array model or GLAM was introduced in 2006. Such models provide a structure and a computational procedure for fitting generalized linear models or GLMs whose model matrix can be written as a Kronecker product and whose data can be written as an array. In a large GLM, the GLAM approach gives very substantial savings in both storage and computational time over the usual GLM algorithm.
Suppose that the data is arranged in a -dimensional array with size ; thus,the corresponding data vector has size . Suppose also that the design matrix is of the form
The standard analysis of a GLM with data vector and design matrix proceeds by repeated evaluation of the scoring algorithm
where represents the approximate solution of , and is the improved value of it; is the diagonal weight matrix with elements
is the working variable.
Computationally, GLAM provides array algorithms to calculate the linear predictor,
and the weighted inner product
without evaluation of the model matrix
In 2 dimensions, let then the linear predictor is written where is the matrix of coefficients; the weighted inner product is obtained from and is the matrix of weights; here is the row tensor function of the matrix given by
where means element by element multiplication and is a vector of 1's of length .
These low storage high speed formulae extend to -dimensions.
GLAM is designed to be used in -dimensional smoothing problems where the data are arranged in an array and the smoothing matrix is constructed as a Kronecker product of one-dimensional smoothing matrices.
- Currie, I.D.;Durban, M.;Eilers, P. H. C. (2006) "Generalized linear array models with applications to multidimensional smoothing",Journal of the Royal Statistical Society, 68(2), 259-280.